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# In how many ways can the letters of the word ACCURACY be rearranged so that no two A's appear as consecutive letters in the rearrangement?

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In how many ways can the letters of the word ACCURACY be rearranged so that no two A's appear as consecutive letters in the rearrangement?

So the word accuracy can be arranged 3360 times. I don't know how the consecutive A's would work

Aug 13, 2022

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There are $${8! \over 3! \times 2!} = 3360$$ ways to arrange the letters as you found out already.

Use complementary counting and count the ways that the 2 A's can be next to each other.

If the 2 A's are next to each other, there are 7 spots for them. This gives us $${6! \over 3!} = 120$$ ways to arrange the other digits (There are 3 C's).

So, there are $$7 \times 120 = 840$$ ways for the 2 A's to be next to each other.

Thus, there are $$3360 - 840 = \color{brown}\boxed{2520}$$ ways in which no 2 A's are next to each other.

Aug 13, 2022